Transcript Probability: Part 2 Sampling Distributions

Probability: Part 2
Sampling Distributions
Wed, March 17th 2004
Sampling Distribution
 A theoretical distribution that allows us to
calculate probability of our sample stats
– Can then generalize from sample  pop
Ex) pop of 2,4,6,8
y = 5 (mu = pop mean)
Draw random sample of N=2 from that pop and
get 4 and 6, ybar = 5 (pretty good
representation of pop mean!)…
but if we drew 8 and 8, ybar = 8 (not so good)
The difference betw sample estimate and
population parameter = sampling error
(cont.)
 How much confidence should we have
in our sample estimate of the pop
parameter?
 Sampling distribution – gives
probabilities of all possible sample
values
– Found by taking all possible random
samples of size N from pop, compute their
means plot
example
 Can do this for all possible combinations of
N=2 (w/replacement) and calculate ybar each
time:
ybar
f
2
1
(1 way to get ybar=2, 2 then 2)
3
2
(could pull 2 then 4, or 4 then 2)
4
3
etc…
5
4
6
3
7
2
8
1
…if you plot this distribution 
it is your sampling distribution!
Mean of Sampling Distrib.
 Sampling distribution also has a mean
and std dev:
–  ybar = mean of samp distrib = pop mean
– Standard deviation of samp distrib is
called the standard error:
 ybar =  y / sqrt N
…where  y is standard dev of pop (sigma)
Represents average distance between pop &
sample means
Central Limit Theorem
 As N increases, sampling
distribution has less variability &
looks like a normal curve
 As N increases, mean of samp
distribution = mean of population
 Usually when N> 30 sampling
distrib will be normal
(cont.)
 Given this, we’ll use the sampling
distribution to find out how
probable (or improbable/unusual)
our 1 sample happens to be
– Is it a good representation of the pop
or not? Use probability to determine
 As N increases, standard error
decreases & we’ll be more confident
in our sample estimate
Sample Likelihood
 Use z scores, now to find the likelihood
of a sample mean (rather than an
individual score)
 1st find mean & standard error
For IQ test, what is prob of group of 9 students
has mean >= 112?
Pop mean = 100, y = 15
1st, need samp distrib mean & standard error
(cont.)
 Ybar (m in lab) = 100
 Ybar (x or s in lab) = 15 / sqrt (9) = 5
Z = ybar -  / ybar
Z = 112-100 / 5 = 2.4
Use unit normal table to find probability of
z=2.4, p = .0082
So very unlikely (.0082) to get a sample of
9 students w/average IQ of 112 from pop
with  = 100